Analytical Modeling of Iron Lossesfor a PM Traction Machine

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Degree project in

Analytical Modeling of Iron Losses for a

PM Traction Machine

ALESSANDRO ACQUAVIVA

XR-EE-E2C 2012:002 Master of Science Thesis Stockholm, Sweden 2012

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Analytical Modeling of Iron Losses

for a PM Traction Machine

By

Alessandro Acquaviva

Master of Science Thesis

XR-EE-E2C 2012:002

Royal Institute of Technology Laboratory of Electrical Energy Conversion

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Acknowledgements

This master thesis was completed as collaboration between Bombardier Transportation and the laboratory of Electrical Energy Conversion at KTH.

I would like to thank in first place my supervisor at Bombardier Dr. Florence Meier and Associate Professor Juliette Soulard, KTH, for making this project possible and for guiding me through the whole work.

I am grateful to my parents for financially supporting this wonderful experience abroad; to my mom in particular for always giving me moral support and to my dad for being an example for me through the years.

I would like also to thank my girlfriend for the daily help she gave me.

Last but not least I would like to thank all my friends and my brothers for their support.

Alessandro Acquaviva Stockholm, March 2012

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Abstract

Permanent magnet (PM) machines offer several advantages in traction applications such as high efficiency and high torque per volume ratio. The iron losses in these machines are estimated mostly with empirical laws taken from other types of machines or with finite element simulations (FEM). In the first part of this thesis the objective is to define an accurate analytical model for the stator yoke, teeth and rotor of a PM motor which should work well enough for all operating point (different loads and frequency).

This analytical model is found using an iterative process. After building a loss matrix and flux matrix based on FEM simulations, it is possible to curve fit each of the lines or the rows of the matrix in order to achieve the best fitting for every operating point. This is a very new approach; it was shown that it gives the possibility, even with a very limited number of FEM simulations, to achieve an accurate estimation of the losses.

The second part of this report focuses on optimizing this analytical method, comparing it with other possibilities, analyzing limits and advantages. Special attention is also given to the effects of the losses on the temperatures in different parts of the machine. In the last part of the thesis, the analytical model is used to test a new control strategy. Its goal is to reduce the total losses of the motor and optimize the ratio between torque and total losses for a given driving cycle.

Keywords: PM synchronous motor, Iron losses, Driving Cycle, Curve Fitting, Interpolation, Air-gap Flux, MTPA and MTPW

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Sammanfattning

Permanentmagnetiserade (PM) maskiner erbjuder flera fördelar i traktion applikationer såsom hög verkningsgrad och högt vridmoment per volym. Förluster modeller är viktiga för att kunna dra njuta av maskinens fördelar. Järn förluster i dessa maskiner beräknas oftast med empiriska lagar tagits från andra typer av maskiner eller med finita element simuleringar (FEM). För att börja med utvecklas en analytisk modell för järn förluster i stator rygg, och tänder samt rotorn i en PM motor som bör fungera tillräckligt bra för alla arbetspunkt (olika belastning och frekvens).

Denna analytiska modellen byggs med en iterativ process. En förlust matris och magnetisk luftgapsflöde matris baserade på FEM simuleringar tas fram. Sedan väljs varje linje eller rader i matrisen för att optimera två parametrar i en analytisk uttryck för att uppnå den bäst passande förlust estimering för varje arbetspunkt. Detta är en mycket ny metod och i det här fallet ger en tillräcklig bra uppskattning av förlusterna även med ett mycket begränsat antal FEM simuleringar.

Den andra delen av rapporten fokuserar på förbättring av denna analysmetod, jämförelse med andra möjligheter, och analys av begränsningar och fördelar. Särskild uppmärksamhet ges också till påverkan av förlusterna på temperaturerna i olika delar av maskinen. I den sista delen av avhandlingen används den analytiska modellen för att testa en ny kontrollstrategi. Målet är att minska de totala förlusterna i motorn och optimera förhållandet mellan vridmoment och de totala förlusterna för en given körcykel.

Sökord: Permanentmagnetiserade synkronmotor, järn förluster, körcykel, kurvanpassning, magnetiskt luftgapflöde.

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1. Introduction

Permanent magnet (PM) synchronous motors are getting more and more used in traction applications. The main reason is the higher efficiency compared to induction motor in the same volume. Nowadays efficiency is becoming a very important parameter, since the electric energy demand is increasing and consequently electricity is more expensive.

The cooling of the machine and all the issues connected to it are of primary importance. The rotor is typically the part of the motor that is most difficult to cool. The rotor has theoretically no losses since it is magnetized with magnets (no copper wire current means no Joule losses) and rotates synchronously with the rotating field. Therefore the cooling of the whole motor can be considered less critical than for induction machine.

Typically more than 60% of the motor failures are due to bearings. As a rule of thumb decreasing the temperature of the bearings by 15 °C would double their life time. Most of the heat that is transferred to the bearings comes from the shaft which is directly connected to the rotor. Consequently having theoretically no losses in the rotor is a really relevant advantage for the lifetime of the motor. This is also one of the reasons that can lead to opt for the PM machine instead of induction motor.

Up till now, most of the analytical models for prediction of iron losses for PM machines are taken from

empirical laws, and many of these are derived for induction machines. Equation Chapter (Next) Section 1

The aim of this work is to find an accurate analytical model for the iron losses in the stator yoke, teeth and rotor of a PM motor and to study some control strategy in order to minimize the total losses of the machine. By analytical model is meant that if the iron losses are known for one starting operating point, at a certain frequency and flux, the iron losses of all the other operating points with different frequency and flux can be evaluated by mathematical expressions.

An accurate estimation of the iron losses is needed both for the control and for a correct estimation of the temperatures given by the thermal model. This can lead to a better use of the motor.

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Fig. 1-1 shows ECO4 the new high efficiency technology for trains, developed by Bombardier

Transportation.

The developed iron loss models are based on a comparison with finite element method (FEM) simulation estimation of the losses. This is already a strong limitation. Furthermore, the machine is assumed to be fed by purely sinusoidal currents. This means that all the losses due to inverter switching are not considered. Chapter 2 is a brief overview on the equations and methods used conventionally for evaluating the iron losses in AC machines. The description is focusing on the one implemented in the FEM program Flux [2]. Chapter 3 is an introduction to PM motors with some of the main concepts being introduced. Chapter 4 shows the geometry of the investigated motor and the first model (called starting model). The analysis and discussion of the method used in this master thesis start in this chapter as well. In Chapter 5, the analytical model is explained in details and applied to the stator tooth iron losses. In Chapter 6, the same method is applied to the stator yoke and the rotor. In Chapter 7, the analytical model is tested on a different geometry and with different matrix dimensions. Chapter 8 shows the effect of an error in the evaluation of iron losses on the temperatures in the different parts of the motor through a sensitivity analysis. In Chapter 9, a control strategy that optimizes the total losses utilizing the analytical model developed in the previous chapter is studied. Chapter 10 contains the analysis of the obtained results and the conclusions.

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2. Literature review on iron loss models in AC machines

The aim of this chapter is to give an overview about the models used for predicting iron losses and to focus on the ones implemented in the FEM software used in the simulations. A good overview about the available iron loss models and a comparison between certain models for analytical and numerical machine

design methods is given in [3]. Equation Chapter (Next) Section 1

The base physical phenomenon behind losses in ferromagnetic materials is Joule heating [4]. Every time a change in the magnetization occurs there is a movement of domain walls, this creates eddy currents which turn into Joule heating. Therefore, it is important to keep in mind that to separate the iron losses for example in eddy current losses, hysteresis losses and excess losses is just an empirical approach. This is done to try to separate the different physical influences and relate them to frequency and flux density.

2.1. Overview of iron loss models

During a machine design process in order to predict the iron losses one can choose among a wide range of different iron loss models for electrical machines. It is important also to take into account the effect of the assembly and manufacturing process that can influence the iron losses.

Fig. 2-1 gives an overview of the most often used methods for determining iron losses [3].

Fig. 2-1 Model approaches to determine iron losses in electrical machines [3]

The first group of models is based on Steinmetz equation [5] which is

(2.1) =

Where CSE, α and β are three coefficients determined by fitting the equation to the measurement data. The

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modifications of the Steinmetz equation are used to estimate iron losses for non-sinusoidal flux densities.

is the peak value of the flux density.

The second group is based on the separation of the different physical phenomena that cause the losses. The first model was presented by Jordan in [6] and separates the iron losses in eddy current and hysteresis losses.

(2.2) = + = +

A further improvement to the approach was given by Bertotti in [7] [8] [9] [10] which introduced an

additional loss term to take into account a different contribution of iron losses called excess losses or

anomalous losses and to give a physical description of the loss factor Cexc.

(2.3) = + + = + + . .

Where

(2.4) =

S is the cross sectional area of the lamination sample, G is a dimensionless coefficient and is the electric

conductivity of the lamination. takes into account the grain size and the statistical distribution of the

local coercive fields.

In order to obtain higher accuracy in the prediction of iron losses, it is possible to use some mathematical hysteresis models. This can be done if measurements of full hysteresis curve are available for the investigated material. Some classical hysteresis models are the ones studied by Preisach [11] [12] and Jiles/Atherton [13]. There are also several improvements of these iron loss models in literature that can be applied to steel sheets and even to complete electrical machines.

The range of iron loss models available for electrical machines is very wide; they are designed for different purposes and differ in several aspects. The models that are best suited for fast and rough estimation of iron losses are the Steinmetz equations and the loss separation model. They are the ones typically integrated in FEM simulations since the flux density can be determined for each element of the mesh in the post-processing.

In this master thesis, FEM simulations are used to calculate iron losses and to compare them with the analytical model to be developed. Therefore it is important to know and describe how this FEM simulator works with iron losses.

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2.2. Iron loss determination in FEM simulations

The FEM program utilized to run electromagnetic simulations is Flux [2]. The computation of magnetic losses in Flux is dealt by means of the formulas of Bertotti [14].

The computation of the iron losses in steady state AC magnetic applications is done by means of the volume density of average power

(2.5) = _!" +#

6 + . . % Where

coefficient of losses by hysteresis W s T-2 m-3

coefficient of losses in excess S m-1

conductivity of the material W(T s-1)-1.5m-3

thickness of the lamination m

frequency Hz

peak value of the flux density T

! coefficient of filling -

Table 1 Symbols of equations used in this paragraph

and are typically obtained from manufacturer’s data sheet.

The average power dissipated in a volume region is written as [14]

(2.6) & = '

() *

The computation of the iron losses in transient magnetic applications is done by means of the volume density of the instantaneous power

(2.7) +,- = _!. +# 6 /, +,-0 + /_{, +,-0} . 1 The volume density average power over a period is

(2.8) ₂₃= 1

5 6 +,- ,

7

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13 (2.9) ₂₃ = _!+ _!1 5 6 .# 6 /, +,-0 + /_{, +,-0} . 1 , 7

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3. PM synchronous machines

Historically all the high power industrial or traction motors were induction motors or DC motors with excitation windings. Permanent magnets were typically used for small applications because it was difficult to find a material capable of retaining a high-strength field. Only with the recent advances in material technology that allowed the creation of permanent magnets with high energy density, such as SmCo or NdFeB magnets, it has been possible to

develop compact high-power motors without field coils.

The main advantages that PM machines offer in traction application are high torque per volume ratio. Theoretically no losses in the rotor lead to a higher efficiency of the motor. Fig. 3-1 shows a PM motor of the MITRAC series by Bombardier Transportation.

Equation Chapter (Next) Section 1

3.1. dq-axis representation

The best way to describe a PM machine with equations is the dq-axis representation in a rotating coordinate system. The coordinate system is rotating with the stator flux and therefore is synchronous to the rotor itself in the case of a synchronous motor.

The direct axis (d-axis) lies in the direction of the flux, while the quadrature axis (q-axis) is 90 electrical degrees ahead counter clockwise.

The following matrixes are used to transform all the three phase stator related quantities to a rotor related two phase rotating representation (also known as the Clarke and Park transformations) [15]:

(3.1) 89₉: ;< 7 = /992= 9 0 7 ∗2₃ A B B B C cos+G,- Hsin+G,-cos KG, H2#_{3 L Hsin KG, H}2#_{3 L} cos KG, +2#_{3 L Hsin KG, +}2#_{3 LM}N N N O

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15 (3.2) KP_P: ;L 7 = /PP2= P 0 7 ∗2₃ A B B B C cos+G,- −sin+G,-cos KG, −2#_{3 L −sin KG, −}2#_{3 L} cos KG, +2#_{3 L −sin KG, +}2#_{3 LM}N N N O The factor

Q allows the amplitude of the current and voltage vectors in the new coordinate system to be the

same for the three phase representation; this is called current and voltage invariant method. The Park transform is valid with the following assumptions:

• No magnetic saturation

• Stator windings with a sinusoidal distribution

• No space harmonics in the magneto-motive force distribution

3.2. Dynamic equations

A complete dynamic set of equations that describes the PM synchronous machine is [16]:

(3.3) 9_: = RP_:+ Ψ: , − 2 G( Ψ; (3.4) 9;= RP;+ Ψ_{, +}; _2 G( Ψ: (3.5) Ψ_:= T_:P_:+ Ψ_U (3.6) Ψ_; = T_;P_; (3.7) 5 =3₂_{2 VΨ}:P;− Ψ; P:W (3.8) 5 = X G_{, + 5}( (

3.3. T-ω Diagram

Field-weakening operation is extremely important for PM synchronous motors. The machine is fed with a negative d-axis current in order to de-flux the machine at high speeds. A negative d-current creates a flux in the air-gap that opposes the flux created by the PM’s. This is a way to deal with a limited voltage level applied by the inverter in order to reach higher speeds.

It is possible to plot the maximum torque as a function of the speed of the machine into a torque-speed diagram.

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Fig. 3-2 T-ω Diagram for an ideal motor [17]

The maximum torque can be delivered from the machine until the point in which the maximum power is

reached. The speed corresponding to that point is referred as the base speed of the motor nr. The speed

range between 0 to nr is called constant torque speed range (CTSR). If a higher speed needs to be reached it

is necessary to reduce the torque in order to keep the power constant to the maximum value. The speed

range between nr to nmax is called constant power speed range (CPSR). Fig. 3-2 shows the CPSR and CTSR for

an ideal motor.

3.4. Types of permanent magnet machines

a) b) c) d)

Fig. 3-3 Typologies of PM machines, from left: a) Surface PM synchronous (SPM) machine b) Interior PM (IPM) synchronous

machine c) Surface Inset PM (SIPM) synchronous machine d) Interior PM with circumferential orientation synchronous machine

Fig. 3-3 shows four main types of PM machines. The surface mounted PM offers a simple mechanical construction but poor robustness. Another simple mechanical construction but with a higher robustness is

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17 offered by the surface inset PM motor. This solution has a slightly higher saliency compared to the surface mounted motor. The interior permanent magnet (IPM) synchronous machine is typically used for high speed applications due to the high mechanical robustness. The drawback is that the manufacturing is more complex compared to the other solutions.

The motor that is analyzed in this master thesis is an IPM motor, the geometry and the main characteristics are shown in the next chapter.

3.5. Driving cycles

A series of data points representing the speed and torque versus time of a vehicle is called driving cycle. Driving cycles are produced to assess the performance of vehicle’s motor in various ways. In this master thesis, they are utilized widely, for thermal simulations and to calculate average total losses, iron losses and make comparisons between different analytical models.

Since the most common and useful representation of operating points of the motor is the d-q current plane it becomes very interesting to show which sectors of this plane are being used and which are not with a certain driving cycle.

Depending on the control strategy adopted the same driving cycle can lead to different operating points. Typically an optimal curve is implemented in the control software in order to make the motor work as close as possible to it. A very common control strategy is the maximum torque per ampere (MTPA), more about how to obtain such curve is shown in chapter 9.

The frequency (which is directly proportional to the speed) versus time plot of one of the driving cycles that is utilized widely in this thesis is shown in Fig. 3-4. This is a driving cycle for a regional train.

Fig. 3-4 Frequency as a function of time for one of the utilized driving cycles

0 1000 2000 3000 4000 5000 6000 7000 8000 0 20 40 60 80 100 120 140 160 time [s] fr e q u e n c y [ H z ] Driving cycle:CY1/CY2

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4. Investigated geometry

The motor studied is an IPM synchronous motor with two pole pairs. It has two magnet layers (see Fig. 4-1).

4.1. Motor data

The investigated motor data is showed in Table 2.

Cooling Air cooled self-ventilated

Type of winding Form winding

Number of poles 4

Number of slots 36

Nominal speed 5500 rpm

Max current, peak 700 A

Outer stator diameter 460 mm

Starting torque 3050 Nm

Table 2 Investigated motor data

The geometry of the analyzed motor is shown in Fig. 4-1. Equation Chapter (Next) Section 1

Fig. 4-1 Geometry of the analyzed motor

4.2. Initial analytical model

In the initial analytical model there are three different analytical models for the three main parts of the machine: the stator yoke, the stator tooth and the rotor. These different areas considered for calculating the losses are shown in Fig. 4-2.

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Fig. 4-2 Different parts of the iron of the motor: rotor (light blue), stator teeth (grey) and stator yoke (blue)

The objective of this analytical model is to derive in the most accurate way the iron losses of the motor knowing the flux maps, the currents, the geometry of the machine and the iron losses calculated with FEM simulations at only one single operating point. It is necessary to have the value (or a good approximation) of the flux for every operating point. This means that it is possible to create flux matrixes (maps) for both the flux in the d and in the q direction with rows and columns corresponding to Id and Iq axis.

One of the important characteristics of this analytical model is that it has to be as simple as possible, since it has to be implemented both in the control and for a thermal analysis of the motor.

In the given model there are two variables, the airgap flux and the frequency. The airgap flux for each

operating point Ψ_Y is calculated as follows:

(4.1) Ψ_Y = ZΨ_:+ [Ψ_;H T_\+P_:+ [P_;-Z

Fig. 4-3 shows the flux in the airgap for different id and iq. The second plot with respect to iq shows clearly

the non-linear characteristic of the machine. Non-linearity created by flux saturation in this case is the reason why it is very difficult to find an accurate analytical model.

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Fig. 4-3 Dependency of the flux in the airgap as a function of id and iq

Equations 4.2, 4.3 and 4.4 are the three parts of the initial analytical model. This is an empirical model inspired by the one used for induction motors.

Iron losses in stator teeth:

(4.2) &_{!_} = &&^_5 ∗ 100 + 100 + a∗ a∗ ΨY Ψ_Y(

Iron losses in rotor:

-7000 -600 -500 -400 -300 -200 -100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 id [A] Flux in airgap vs Id F lu x a ir g a p /N o m in a l fl u x 0 100 200 300 400 500 600 700 0 0.2 0.4 0.6 0.8 1 1.2 1.4 iq [A] Flux in airgap vs Iq F lu x a ir g a p /N o m in a l fl u x Increasing iq Increasing |id|

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21 (4.3) &_{!_(b} = &&^_Rc ∗ K aL ∗_ΨΨY Y(

Iron losses in stator yoke:

(4.4) &_! defg= &&^_R ∗ 100 + 100 + a∗ a ∗ h ΨY Ψ_Y(i

All these equations have a term with a quadratic dependency from the frequency. The equations corresponding to the stator have also a term with a linear dependency to the frequency. Since the quadratic dependency to the frequency is typical of eddy current losses and the linear dependency of hysteresis losses, it is possible to conclude that in the model used the rotor hysteresis losses are neglected. This can be reasonable because the iron losses in the rotor are mainly caused by the flux harmonics due to the slotting effect which are at a frequency multiple of the fundamental.

A very important step to achieve an accurate estimation of the iron losses is choosing the reference operating point. From the losses at this point, all the others will be calculated. It defines the value of the loss coefficients PPFET, PPFER and PPFERO. The values of the iron losses, airgap flux and leakage inductance for the reference operating point are presented in Table 3.

This reference point is chosen such that it can be considered in the middle of the operating area of some typical trains driving cycles.

The chosen reference operating point is:

(4.5) P_:_j= −150 l, P_;_j= 150 l, _a= 90 oz

For this operating point the coefficients can be calculated running a FEM simulation. The values are given in Table 3. Ψ_Y( 2.91 pq &&^_5 1710 p &&^_R 2320 p &&^_Rc 197 p Ts)U 160.584 vo

Table 3 Values of the coefficients calculated for the reference operating point

To have a global view on how the analytical model is performing it would be necessary to build a full map of iron losses in all the possible operating points of the machine. Since there are a large number of possible

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operating points, in order to have an acceptable view, 225 simulations are run. This means that a 15x15 matrix can be built for each of the iron losses in the stator teeth, stator yoke and rotor.

The chosen frequencies are: 50 Hz, 90Hz which is the nominal frequency of the motor and 175 Hz which is the frequency of the machine operating at the maximum speed. The 225 simulations for each frequency are chosen such that: Id goes from 0 to -700 A, and Iq goes from 0 to 700 A.

Fig. 4-4 shows the comparison between the initial analytical model and the FEM simulations. The losses are shown only for the nominal frequency.

From Fig. 4-4 it is clear that this model needs to be improved. The losses in the yoke are well predicted while in the other parts of the machine there are significant differences. The main problem in dealing with the dependency to the flux is saturation which is not considered properly in the initial model.

Fig. 4-4 Iron losses obtained with analytical models from induction motor and iron losses obtained with a FEM simulation for

different parts of the machine at 90 Hz

-7000 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 id [A] L o s s [ W ]

Iron losses tooth 90 Hz

Analytical model FEM -7000 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 3500 4000 4500 id [A] L o s s [ W ]

Iron losses yoke 90 Hz

Analytical model FEM -7000 -600 -500 -400 -300 -200 -100 0 50 100 150 200 250 300 id [A] L o s s [ W ]

Iron losses rotor 90 Hz

Analytical model FEM

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5. Analytical modeling of iron losses in stator tooth of a PM

machine

In this part the focus is shifted on the stator teeth iron losses .The losses at a given frequency obtained

from FEM being dependent on id, iq, it would be necessary to do a surface fitting to develop a suitable

analytical model. However, since this is a complex tool and it is hard to set the starting equations in order

to make the tool work, a multiple curve fitting is chosen instead. Equation Chapter (Next) Section 1

5.1. Curve fitting of losses in stator tooth with i

q

constant (MODEL 1)

As said in the previous chapter a 15x15 matrix is built containing the iron losses in the stator teeth

calculated running 15x15=225 FEM simulations at 90 Hz. Each row of the matrix has a different iq value

starting from the 1st equal to 0 A to the 15th equal to 700 A. Each column of the matrix has a different id

value starting from the 1st equal to 0 A to the 15th equal to -700 A as shown in Fig. 5-1. The value of 700 A is

the maximum current for the motor.

Fig. 5-1 Matrix built for curve fitting of losses in stator tooth at one frequency

The curve fitting is done for each of the 15 rows of the matrix; this means that each curve fitting is done for

a constant value of iq. The main interest is to look at the dependency of the losses in the iron tooth as a

function of the flux in the airgap. To do this the curve fitting of the stator tooth iron losses is done as

function of the flux in the airgap when the id current is changing, because as said above the iq is constant for

each row.

The curve fitting general equation used is (the reference point and the corresponding coefficients are kept as in chapter 4):

(5.1) w = &&^_5 ∗ x ∗ y=

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(5.2) y = ΨY

Ψ_Y(

For each of these curve fittings the coefficients a and b are calculated. This means that for each value of id

there is a different value of the two coefficients a and b.

According to Bertotti’s equation of iron losses assuming the excess losses can be neglected, the iron losses depend on the frequency with a linear and a quadratic term that correspond respectively to hysteresis losses and eddy current losses.

The model used is:

(5.3) &_{!_}= &&^_5 ∗ x ∗ 100 + 100 + a∗ a∗ h ΨY Ψ_Y(i =

The values of a and b calculated as a function of id are shown in Fig. 5-2.

Fig. 5-2 Values of the of the stator tooth fitting coefficients for different values of iq

Fig. 5-2 shows that the reference operating point being chosen is fairly good since the value of a is close to 1. 0 100 200 300 400 500 600 700 -4 -3 -2 -1 0 1 2 iq [A] F it ti n g c o e ff ic ie n ts model: y=PPFET*a*xb a b

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The fundamental flux in the airgap decreases with increasing negative id as shown in Fig. 4-3. A negative

flux exponent b means that the iron losses in the stator teeth are increasing even though the fundamental flux density is decreasing.

To explain this negative exponent, it is necessary to analyze what happens in the stator tooth when the

value of id is increasing (i.e. more negative). This is done running a FEM simulation at different values of id

keeping iq constant. The flux density waveform in the middle of the stator tooth is plotted for two different

values of id, the first one is run with id = -150 A and the second with id =-550 A with iq constant to 150A. This

is shown in Fig. 5-3, it is interesting to look at the spectrum of the two waveforms. The fundamental

decreases with increasing id as expected from Fig. 4-3 but the harmonic distortion is increased.

The increase of high order harmonics as the 13th and the 15th but also the lower ones is the reason why the

losses are increasing even though the machine is getting de-fluxed.

Fig. 5-3 Comparison between flux density in the middle of the stator tooth with id = -150 A (left) and with id =-550 A (right) iq is

constant to 150A

Another important consideration is that the considered current values are in a much wider range than the practical ones. In the driving cycles provided for this motor, the Id current would never reach the value of 500 A because the magnets could get damaged in such case. This is clearly shown in Fig. 4-3, the flux from

the magnets is totally canceled for a value of id around -500 A.

5.2. Curve fitting of losses in stator tooth with i

d

constant (MODEL 2)

The same process can be done for constant values of id. This means that the curve fitting is done for each of

the 15 columns of the matrix shown in Fig. 5-1. The curve fitting general equation and the model for iron

losses in the stator tooth are the same as in section< 5.1. The values of a and b calculated as a function of id

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Fig. 5-4 Values of the stator tooth fitting coefficients for different values of id

Fig. 5-4 shows that the choice of the reference operating point is fairly good since the value of a is close to 1 as in the previous curve fitting. This time the flux exponent starts as positive and is decreasing with

increasing negative id. The fundamental flux in the airgap increases with increasing iq as shown in Fig. 4-3.

This explains the positive flux exponent for low values of id, because in this case the iron losses in the stator

tooth are increasing if the flux density fundamental is increasing as shown in Fig. 5-5.

Fig. 5-5 Comparison between flux density in the middle of the stator tooth with iq = 150 A (left) and with iq =550 A (right) id is

constant to -150A -700 -600 -500 -400 -300 -200 -100 0 -0.5 0 0.5 1 1.5 2 2.5 id [A] F it ti n g c o e ff ic ie n ts model: y=PPFET*a*xb a b

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5.3. Analytic model of fitting coefficients

It is not possible or anyway it is very unpractical to run simulations and curve fit the map losses for each different motor and every possible operating point to curve fit loss maps. To avoid this, an analytical model of the variation of a and b coefficients calculated in the previous two chapters is looked for. As said previously one of the objectives is to have it as simple as possible, therefore the lowest general equation order that approximates the coefficients should be chosen.

Fig. 5-6 shows the analytical model found for the coefficients shown in Fig. 5-2. To fit properly these coefficients a cubic equation has to be used for the b coefficient and to approximate well the a coefficient

even a higher order equation has to be used, a 5th order general equation. This is a big restriction for

MODEL 1 since it would make the final analytical model for iron losses in the stator tooth complex.

Fig. 5-6 Analytical models of the fitting coefficients for different values of iq

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 iq/imax a a vs. id fit 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 iq/imax b b vs. id fit 2

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The general equations used to fit the coefficients are: For the b coefficient

(5.4) q+y- = ∗ yQ + ∗ y + _Q∗ y + _z where (5.5) y = K P; PU2L and (5.6) P_U2 = 700 l = 22.09 = 41.31 Q= 20.66 z= −0.1962

For the a coefficient

(5.7) x+y- = ∗ y + ∗ yz + _Q∗ yQ + _z∗ y+ ∗ y + _{

where

= 45.38 = 137.2 Q= 147.5 z= 63.99 = 7.124 {= 0.4771

Fig. 5-7 shows the analytical model found for the coefficients shown in Fig. 5-4. In MODEL 2 the plots of the two coefficients are much simpler and the order of the general equation used to approximate them can be lowered to a linear model for coefficient a and a quadratic model for coefficient b. This is a really big advantage of MODEL 2 because it reduces the complexity of the final equation used for the iron losses in the stator teeth.

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29

Fig. 5-7 Analytical models of the fitting coefficients for different values of id

(5.8) q+y- = ∗ y + ∗ y + _Q where (5.9) y = K P: PU2L and -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.5 0 0.5 1 1.5 2 2.5 id/imax b b vs. id fit 2 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 id/imax a a vs. id fit 1

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= 2.860 = 5.037 Q= 2.105

(5.10) x+y- = ∗ y +

where

= −0.697 = 0.578

Now that an analytical model is defined, the iron loss model in the stator tooth would be:

(5.11) &_{!_} = &&^_5 ∗ x+y- ∗ 100 +

100 + a∗ a∗ h ΨY Ψ_Y(i

=+-5.4.

Stator tooth iron losses with the new models

The loss percentage error can be calculated and used to evaluate how close to the FEM and the analytical models are to each other. It is is defined as follows:

(5.12) Δ&_!% =&!~− &!

&! ∗ 100

By using MODEL 1 at the three different values of frequency the loss percentage error is obtained. Fig. 5-8, Fig. 5-9 and Fig. 5-10 show respectively the loss percentage error and the losses calculated both with the analytical model and with the FEM simulation at 50Hz at 90Hz and at 175Hz. The improvement compared to the starting model is remarkable. From the plots it is possible to confirm that the dependency from the frequency is reliable. The loss percentage error plot with the color scale is a fast and easy way to understand which operating points on the current plane are more or less critical for the model (for critical is meant a high absolute value of the loss percentage error).

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31

Fig. 5-8 Loss percentage error and iron losses in the stator tooth calculated both with the analytical model and with the FEM

simulation at 50Hz of MODEL 1

Fig. 5-11, Fig. 5-12and Fig. 5-13 show respectively the loss percentage error and the losses calculated both with the analytical model and with the FEM simulation at 50Hz at 90Hz and at 175Hz using MODEL 2.

-700 -600 -500 -400 -300 -200 -100 0 200 300 400 500 600 700 800 900 1000 1100 id [A] Iron Losses in tooth 50 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth iq /i m a x -40 -20 0 20 40 60 80 -7000 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 id [A] Iron Losses in tooth 90 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth iq /i m a x -40 -20 0 20 40 60 80 -700 -600 -500 -400 -300 -200 -100 0 1000 2000 3000 4000 5000 6000 7000 8000 id [A] Iron Losses in tooth 175 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth 175 Hz iq /i m a x -40 -20 0 20 40 60 80

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Both the models are fitting the losses much better than the initial model, but it is now interesting to understand which one should be used and why.

-700 -600 -500 -400 -300 -200 -100 0 200 300 400 500 600 700 800 900 1000 1100 1200 id [A] Iron Losses in tooth 50 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth iq /i m a x -40 -20 0 20 40 60 80 -700 -600 -500 -400 -300 -200 -100 0 200 300 400 500 600 700 800 900 1000 1100 1200 id [A] Iron Losses in tooth 50 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth iq /i m a x -40 -20 0 20 40 60 80 -700 -600 -500 -400 -300 -200 -100 0 1000 2000 3000 4000 5000 6000 7000 8000 id [A] Iron Losses in tooth 175 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth 175 Hz iq /i m a x -40 -20 0 20 40 60 80

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33

5.5. Comparison between MODEL 1 and MODEL 2 for evaluating iron

losses in stator tooth

In order to evaluate which of the two models is better the driving cycle shown in chapter 3 is considered, both at full DC-link voltage 1650 V, named CY1, and with reduced DC-link voltage 1250 V, CY2. It is interesting to check both possibilities because with low voltage the operating points tend to “leave” the maximum torque per ampere (MTPA) curve at a lower speed.

It is necessary for each time step of the driving cycle to check frequency, fluxes and currents and calculate the iron losses. Therefore an interpolation between different frequencies is needed.

The simulations and the models are run for four different frequencies: 0, 50, 90, 175 Hz. (at 0 Hz the losses in the motor are supposed to be 0).

The function interpolates four different loss maps (one for each frequency) for positive and negative iq

(-700 A to (-700 A) and negative id (-700 A to 0 A). The loss map for negative values of iq is exactly the

symmetrical with respect to iq =0 to the positive part. For each point of the driving cycle the speed, as well

as the values of id and iq are checked in order to interpolate for both current and frequency and find the

loss percentage error between the FEM simulation and the model used. The mean loss error and the mean electrical frequency of the motor are also calculated for each driving cycle.

A comparison between linear and cubic interpolation is shown in Fig. 5-14. Since the difference is very small while the computation time is much longer (more than 10 times longer) for the cubic interpolation, a linear interpolation is prefered. The method used is a 3D linear interpolation, i.e. a linear interpolation for each variable id and iq and f.

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Fig. 5-14 Comparison between linear and cubic interpolation

Once the interpolation is done it is possible to plot the losses for each time step of the driving cycle for both models.

The results of these interpolations are shown in Appendix A. Fig.A.1, Fig.A.2, Fig.A.3 and Fig.A.4 show the results of this interpolation applied to MODEL 1 for the two different driving cycles and Fig.A.5, Fig.A.6, Fig.A.7 and Fig.A.8 show the same thing for MODEL 2.

If a global look is given to Appendix A it can be said that both the models are quite accurate on average, MODEL 1 is slightly more accurate than MODEL 2 on the whole driving cycle for both CY1 with full voltage and CY2 with reduced voltage 1250 V. However, it is also important to notice that MODEL 1 has some operating points in which the loss percentage error is above 80% while MODEL 2 is always under 20%, so it can be concluded that MODEL 2 is more reliable and less cycle dependent than MODEL 1.

It is also really interesting to represent in the d-q current plane an iso-value map of the loss percentage error and the total current showing also operating points for both driving cycles. The operating points of the driving cycles are split between the three different frequencies, from 0 to 70 Hz the points are plotted in the 50 Hz figure, from 70 to 130 Hz the points are plotted in the 90 Hz figure and for frequencies above 130 Hz they are plotted in the 175 Hz. This is done in order to view where the operating points are on the d-q current plane for different frequencies, but also to evaluate if some of the operating points lay where the loss percentage error is high.

0 50 100 150 200 250 300 350 400 450 500 -40 -20 0 20 40 60 80 100 120 time [s] d P %

Comparison between linear and cubic interpolation for driving cycles time plots linear cubic

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35 The iso-value maps are shown in Appendix B. Fig.B.1, Fig.B.2 and Fig.B.3 show respectively the iso-value maps for 50, 90 and 175 Hz for MODEL 1. Fig.B.4, Fig.B.5 and Fig.B.6 show respectively the iso-value maps for 50, 90 and 175 Hz for MODEL 2.

Appendix B gives a useful view of how the operating points of the driving cycle are situated in the d-q current plane, and from these plots it can be seen that the areas of the d-q current plane with high values of loss percentage error are “almost empty”. This is especially true for MODEL 2. That is why the second model can be considered more reliable.

From the considerations done so far MODEL 2 is chosen as the best one and the two main reasons are: more reliable and simpler model of the equations.

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6. Analytical modeling of iron losses in rotor and stator yoke of a

PM machine

The same procedure used in chapter 2 can be used to fix an analytical model both for the rotor and the

yoke iron losses. Equation Chapter (Next) Section 1

6.1. Rotor fitting with MODEL 2

From similar considerations from the previous chapter, more reliable and much simpler equations due to easier curve fitting of the coefficients, it has been chosen to apply MODEL 2 also to the rotor.

Fig. 6-1 shows the analytical models of the fitting coefficients for different values of id.

Fig. 6-1 Analytical models of the rotor fitting coefficients for different values of id

The general equations used to fit the coefficients are:

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 id/imax a a vs. id fit 1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.5 0 0.5 1 1.5 2 2.5 id/imax b b vs. id fit 2

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37 For the b coefficient

(6.1) q+y- = ∗ y + ∗ y + _Q where (6.2) y = K P: PU2L and = 1.462 = 4.077 Q= 2.288

(6.3) x+y- = ∗ y +

where

= H0.397 = 0.287

Now that an analytical model is defined, the iron loss model in the rotor would be:

(6.4) &_{!_(b} = &&^_Rc ∗ x+y- ∗ K

aL

∗ h_ΨΨY

Y(i

=+-Fig. 6-2, =+-Fig. 6-3 and =+-Fig. 6-4 show respectively the loss percentage error and the losses calculated both with the analytical model and with the FEM simulation at 50Hz at 90Hz and at 175Hz.

Fig. 6-2 Loss percentage error and iron losses in the rotor calculated both with the analytical model and with the FEM simulation

at 90Hz of MODEL 2 -7000 -600 -500 -400 -300 -200 -100 0 50 100 150 200 250 id [A] Iron Losses in rotor 90 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% rotor iq /i m a x -30 -25 -20 -15 -10 -5 0 5 10 15 20

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at 50Hz of MODEL 2

at 175Hz of MODEL 2

The dependency to the frequency is the major discrepancy in this model, the fact that at 50 Hz the losses are almost everywhere underestimated and at 175 Hz overestimated suggests a lower grade dependency to the frequency than the model that has been chosen.

6.2. Stator yoke fitting with MODEL 2

The exact same model can be applied to the stator yoke even though the starting model of the yoke is working well.

For the same reasons as above it has been chosen to apply MODEL 2 also to the stator yoke.

Fig. 6-1Fig. 6-5 shows the analytical models of the fitting coefficients for different values of id.

-7000 -600 -500 -400 -300 -200 -100 0 10 20 30 40 50 60 70 id [A] Iron Losses in rotor 50 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% rotor iq /i m a x -40 -35 -30 -25 -20 -15 -10 -5 0 5 -7000 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700 800 900 id [A] Iron Losses in rotor 175 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 dP% rotor 175 Hz id/imax iq /i m a x -10 0 10 20 30 40

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39

Fig. 6-5 Analytical models of the stator yoke fitting coefficients for different values of id

(6.5) q+y- = ∗ y + ∗ y + _Q where (6.6) y = K P: PU2L -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 id/imax a a vs. id fit 1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.8 1 1.2 1.4 1.6 1.8 2 id/imax b b vs. id fit 2

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and

= −0.609 = 0.575 Q= 1.906

(6.7) x+y- = ∗ y +

where

= H0.354 = 0.965

Now that an analytical model is defined, the iron loss model in the rotor would be:

(6.8) &_{!_b} = &&^_R ∗ x+y- ∗ K

aL

∗ h_ΨΨY

Y(i

=+-Fig. 6-6, =+-Fig. 6-7 and =+-Fig. 6-8 show respectively the loss percentage error and the losses calculated both with the analytical model and with the FEM simulation at 50Hz at 90Hz and at 175Hz.

Fig. 6-6 Loss percentage error and iron losses in the stator yoke calculated both with the analytical model and with the FEM

simulation at 90Hz of MODEL 2 -7000 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 3500 4000 4500 id [A] Iron Losses in yoke 90 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/max dP% yoke iq /m a x -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10

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41

As in the case of the rotor also here there is a small error in the dependency to the frequency. Anyway it is a minor problem since the loss percentage error in the utilized region of the current plane is still very small, the drawback of increasing the complexity of the analytical model by changing also the frequency exponent makes the correction of this error not worth it.

-7000 -600 -500 -400 -300 -200 -100 0 200 400 600 800 1000 1200 1400 1600 1800 2000 id [A] Iron Losses in yoke 50 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 dP% yoke id/imax iq /i m a x -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 -7000 -600 -500 -400 -300 -200 -100 0 2000 4000 6000 8000 10000 12000 id [A] Iron Losses in yoke 175 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% yoke iq /i m a x -80 -60 -40 -20 0 20

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7. Analytical model tested on a different geometry and lower limit

of matrix size

The final objective of this work is to find a general model for PM machines. So far the models of the different parts of the machine are an optimized analytical model that describes quite accurately the iron losses for a given geometry. Therefore, the models are tested with modified motor geometries.

7.1. Test on different geometry

The main objective of this chapter is to test the different models found in the previous chapters with new

geometries. Equation Chapter (Next) Section 1

The main reason to increase the complexity of the iron loss models was to take into account flux density distortion created by the slot leakage at flux-weakening and non-linear effects as saturation. The first choice in modifying the geometry is to increase the stator tooth width (more slot leakage) and decrease the stator yoke height. The stator slot area is kept constant.

The new geometry is shown in Fig. 7-1.

Fig. 7-1 Modification of the geometry

The model that is used for the stator tooth iron losses is MODEL 2 and the results for 90 Hz are shown in Fig. 7-2 and Fig. 7-3. All the losses and the coefficients a, b are recalculated at 90 Hz for both the yoke and the tooth. The rotor geometry is unchanged. The two figures show that the analytical model is still fitting properly for this new geometry even though the flux saturation is very high in the yoke.

This is not a proof that the method works independent from the geometry, since many approximations were done considering the initial geometry. It could be interesting to see what the limit of validity for those assumptions is. At this stage the model has still the potential to be used for a wide range of machines.

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43

simulation at 90Hz of MODEL 2 for the new modified geometry

7.2. Lower limit of matrix size

Simulating 225 times on FEM for each needed frequency to obtain a 15x15 matrix a procedure that takes a lot of time and creates a lot of data; this could also be a limit.

It is interesting to try to decrease the size of the matrix used to calculate the coefficients for the analytical model. To do this the same method is used with a 8x8 and 4x4 matrix.

The main problem of reducing the size of the matrix is that the approximation of the coefficients could become critical, i.e. the number of points could be too low to make an accurate and significant curve fitting of a and b.

Fig. 7-4 and Fig. 7-5 show how the two coefficients are affected by the matrix reduction in the case of the stator tooth iron loss analytical model at 90 Hz.

-700 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 3500 4000 id [A] Iron Losses in tooth 90 Hz

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth iq /i m a x -40 -30 -20 -10 0 10 -7000 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 Id Current [A] L o s s [ W ]

Iron Losses in yoke 90 Hz

Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 dP% yoke Id Iq -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10

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Fig. 7-4 Coefficient a for the stator tooth at 90 Hz for different matrix dimensions

Fig. 7-5 Coefficient b for the stator tooth at 90 Hz for different matrix dimensions

In Fig. 7-4 and Fig. 7-5 the coefficients corresponding to the 8x8 matrix give almost the same result as the 15x15 while the ones corresponding to the 4x4 matrix are slightly shifted. It is interesting to check how this difference in the coefficients affects the loss percentage error.

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

Matrix dimension comparison, coeff. a

id/imax a 15x15 8x8 4x4 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.5 0 0.5 1 1.5 2 2.5

Matrix dimension comparison, coeff. b

id/imax

b

15x15 8x8 4x4

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45

simulation at 90Hz of MODEL 2 with a 8x8 matrix

simulation at 90Hz of MODEL 2 with a 4x4 matrix

The error obtained with the 15x15 matrix is shown in Fig. 5-12 and as expected it is approximately the same as the one shown in Fig. 7-6 corresponding to the 8x8 matrix.

Comparing Fig. 7-6 and Fig. 7-7 it is possible to say that there is a difference in the loss percentage error but it is not significant. The result obtained with the 4x4 is still much better than the initial model and not far at all from a model obtained with a larger matrix. This is a very important result since it is possible to make such a model with just 16 FEM simulations and obtain an accurate enough estimation of the iron losses.

-7000 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 id [A] Iron Losses in tooth 90 Hz 8x8

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 id/imax dP% tooth 8x8 iq /i m a x -10 0 10 20 30 40 -7000 -600 -500 -400 -300 -200 -100 0 500 1000 1500 2000 2500 3000 id [A] Iron Losses in tooth 90 Hz 4x4

W Analytical model FEM -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 dP% tooth 4x4 id/imax iq /i m a x -30 -25 -20 -15 -10 -5 0 5 10 15 20

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8. Sensitivity analysis

A sensitivity analysis is interesting to evaluate how much an error in the estimation of iron losses affects the temperature of the motor. This is done using the new analytical model presented in the previous chapters.

The new analytical model of the iron losses found in the previous chapter is implemented in a lumped parameter thermal model for traction motors. The thermal model is partly described in [18]. The temperatures in different parts of the motor, stator yoke and teeth, windings, magnets and bearing can be calculated for different driving cycles. For each requested torque, speed and DC-link voltage, the motor current, fluxes and losses are calculated. These parameters are used then to solve the thermal equations of

the lumped parameter model and obtain the temperatures. Equation Chapter (Next) Section 1

Four different thermal simulations are run. The first one is using the loss coefficients found for the reference operating point. The three other simulations are run changing each time only one of them, decreasing it by 20%. The driving cycle used is CY1 the one at full voltage utilized in the previous chapters.

&&^_5 1985 p &&^_R 2798 p &&^_Rc 200 p

Table 4 Reference operating point iron loss coefficients

The same thermal simulation is run reducing by 20% one of the reference operating point iron loss coefficients. It is then possible to calculate the temperature percentage error for each time step and for each part of the motor. The temperature percentage is defined as:

(8.2) Δ_%=−

∗ 100

Where is the temperature calculated for the original values of the iron loss coefficients and is the one

calculated for one of the coefficients with 20% variation.

8.1. Temperature differences

In this paragraph the effect of how much an error in the estimation of the iron losses in the different parts affect the temperatures of the different of the motor is studied for a given driving cycle.

The coefficients used for the stator tooth are shown in Table 5 and have the same values as for reference operating point except PPFET is decreased by 20%. This is a preliminary step to calculate later on the sensitivity coefficients for the different parts of the motor.

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47 &&^_5 1588 p

&&^_R 2798 p &&^_Rc 200 p

Table 5 Coefficients used to evaluate the temperatures for an error of stator tooth iron losses

Fig. 8-1 shows Δ = H , the stator and rotor temperature differences with 20% ΔP iron loss in stator

tooth.

Fig. 8-1 Stator and rotor temperature differences with 20% ΔP iron loss in stator tooth

The plot on the left in Fig. 8-1 is showing the temperature difference for the three main parts of the stator, while the one on the right is showing the temperature error for the PMs and the bearings. The temperature difference in Fig. 8-1 at steady state is quite relevant, in the PMs the coil and the tooth is even above 5 °C. The second analysis is run with the coefficients shown in Table 6, these are the same coefficients of the reference operating point with the PPFER decreased by 20%.

&&^_5 1985 p &&^_R 2238 p &&^_Rc 200 p

Table 6 Coefficients used to evaluate the temperatures for an error of stator yoke iron losses

Fig. 8-2 shows the stator and rotor temperature difference with 20% ΔP iron loss in stator yoke.

0 2 4 6 8 10 12 0 1 2 3 4 5 6 Time [h] T e m p e ra tu re s e rr o r [ °C ]

Stator temperature errors

Coil Tooth Yoke 0 2 4 6 8 10 12 0 1 2 3 4 5 6 Time [h] T e m p e ra tu re s e rr o r [°C ]

Rotor temperature errors

PM Bearing

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Fig. 8-2 Stator and rotor temperature errors with 20% ΔP iron loss in stator yoke

The third analysis is run using the loss coefficients in Table 7 with PPFERO decreased by 20%. &&^_5 1985 p

&&^_R 2798 p &&^_Rc 160 p

Table 7 Coefficients used to evaluate the temperatures for an error of rotor iron losses

Fig. 8-3 shows the stator and rotor temperature difference with 20% ΔP iron loss in stator rotor.

Fig. 8-3 Stator and rotor temperature errors with 20% ΔP iron loss in rotor

It is now possible to calculate the sensitivity coefficients.

8.2. Sensitivity coefficients

The sensitivity coefficients are calculated as:

(8.3) =Δ% Δ&% 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time [h] T e m p e ra tu re s e rr o r [°C ]

Coil Tooth Yoke 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 3.5 4 Time [h] T e m p e ra tu re s e rr o r [°C ]

PM Bearing 0 2 4 6 8 10 12 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time [h] T e m p e ra tu re s e rr o r [°C ]

Coil Tooth Yoke 0 2 4 6 8 10 12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time [h] T e m p e ra tu re s e rr o r [°C ]

PM Bearing

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49 These coefficients are a good indicator of how much an error in one of the iron loss models effects the estimation of the temperatures in the different parts of the motor.

The values of s shown in Table 8 are obtained using the mean Δ_% over one cycle after four cycles to reach

the steady state.

Table 8 Sensitivity coefficients

The rotor losses are a small part of the losses considered. Even if there is a relatively high error in evaluating these losses the effect on the temperatures is ten times smaller than the same error in percentage in estimating the losses in the stator tooth or yoke.

It is always important to remember that the temperature in the magnets is one of the most critical because it can lead to damaging the PMs.

Temp. error Coil Temp. error Tooth Temp. error Yoke Temp. error PM T. error Bearing Loss error Rotor 0,0112 0,0116 0,0106 0,0259 0,0153 Loss error Tooth 0,1759 0,1857 0,1697 0,1791 0,1526 Loss error Yoke 0,1299 0,1372 0,1555 0,1311 0,1257

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9. Maximum torque per watt control

One of the common control strategies utilized for PM machines is the maximum torque per ampere (MTPA) strategy. The principle is to always try to make the motor work with the maximum torque for each value of

current. Equation Chapter (Next) Section 1

9.1. MTPA

The MTPA curve can then be obtained from the definition, for each step of growing current amplitude the current angle that gives the maximum torque is taken. As said in chapter 4 the flux maps are known from

the FEM simulations. This means that the matrixes Ψ_: and Ψ_; as a function of Id and Iq are known.

Since

(9.1) P_:= H sin

(9.2) P_; = cos

Where is the current angle. It is easy to obtain

(9.3) I_:= i_:, ⋯ i_:, ⋮ ⋱ ⋮ i_:_~_, ⋯ i_:_~_, (9.4) I_; = i_;, ⋯ i_;, ⋮ ⋱ ⋮ i_; ~, ⋯ i;~,

It is then possible to obtain the two flux matrixes by linear interpolation from FEM simulation results:

(9.5) Ψ_:= Ψ_:, ⋯ Ψ_:, ⋮ ⋱ ⋮ Ψ_:_~_, ⋯ Ψ_:_~_, (9.6) Ψ_; = Ψ_;, ⋯ Ψ_;, ⋮ ⋱ ⋮ Ψ_; ~, ⋯ Ψ;~,

Analytical Modeling of Iron Lossesfor a PM Traction Machine

Analytical Modeling of Iron Losses for a

PM Traction Machine

ALESSANDRO ACQUAVIVA

Analytical Modeling of Iron Losses

for a PM Traction Machine

By

Alessandro Acquaviva

Master of Science Thesis

Acknowledgements

Abstract

Sammanfattning

Table of contents

1.

Introduction

2.

Literature review on iron loss models in AC machines

2.1.

Overview of iron loss models

2.2.

Iron loss determination in FEM simulations

3.

PM synchronous machines

3.1.

dq-axis representation

3.2.

Dynamic equations

3.3.

T-ω Diagram

3.4.

Types of permanent magnet machines

3.5.

Driving cycles

4.

Investigated geometry

4.1.

Motor data

4.2.

Initial analytical model

5.

Analytical modeling of iron losses in stator tooth of a PM

machine

5.1.

Curve fitting of losses in stator tooth with i

constant (MODEL 1)

5.2.

Curve fitting of losses in stator tooth with i

constant (MODEL 2)

5.3.

Analytic model of fitting coefficients

=+-5.4.

Stator tooth iron losses with the new models

5.5.

Comparison between MODEL 1 and MODEL 2 for evaluating iron

losses in stator tooth

6.

Analytical modeling of iron losses in rotor and stator yoke of a

PM machine

6.1.

Rotor fitting with MODEL 2

6.2.

Stator yoke fitting with MODEL 2

7.

Analytical model tested on a different geometry and lower limit

of matrix size

7.1.

Test on different geometry

7.2.

Lower limit of matrix size

8.

Sensitivity analysis

8.1.

Temperature differences

8.2.

Sensitivity coefficients

9.

Maximum torque per watt control

9.1.

MTPA

=+-5.4.